Question

Help on four questions. How do I solve them?

Answer 1

For 16, check the first picture below.

for 29, check the second picture below.

now, notice, for 29, if we circumscribe a circle around the regular hexagon, each side of the hexagon fits in an equilateral triangle, namely a 60-60-60 triangle, and the apothem, is just an angle bisector of one of those triangles, like in the picture.

since the apothem is an angle bisector, it cuts the central angle in two 30-30 angles, making two 30-60-90 triangles.

using the 30-60-90 rule, and since we know the longer leg, apothem, is 6 units long, then we can get the shorter leg, or "a" in this case.

now, bearing in mind that each side in the hexagon is a+a or 2a long, the perimeter is 12a then, thus,

$\bf \textit{area of a regular polygon}\\\\ A=\cfrac{1}{2}rp\quad \begin{cases} r=apothem\\ p=perimeter\\ ----------\\ r=6\\ p=12a\\ \qquad 12(2\sqrt{3})\\ \qquad 24\sqrt{3} \end{cases}\implies A=\cfrac{1}{2}(6)(24\sqrt{3}) \\\\\\ A=3(24\sqrt{3})\implies A=72\sqrt{3}$

39)

$\bf \textit{arc's length}\\\\ s=\cfrac{\theta \pi r}{180}\quad \begin{cases} r=radius\\ \theta =angle~in\\ \qquad degrees\\ ------\\ r=4\\ \theta =110 \end{cases}\implies s=\cfrac{110\cdot \pi \cdot 4}{180}\implies s=\cfrac{22\pi }{9}$